Question

answer
a reflection over the line y = -x
a counterclockwise rotation of 90° about the origin
a reflection over the line y = x
a counterclockwise rotation of 180° about the origin

Explanation:

Step1: Recall transformation rules

For a reflection over the line $y = -x$, the transformation rule for a point $(x,y)$ is $(-y,-x)$. For a counter - clockwise rotation of $90^{\circ}$ about the origin, the rule is $(-y,x)$. For a reflection over the line $y=x$, the rule is $(y,x)$. For a counter - clockwise rotation of $180^{\circ}$ about the origin, the rule is $(-x,-y)$.

Step2: Analyze the given figure (assume we know the general effect of transformations on shapes)

A reflection over $y = -x$ flips the figure across the line with a negative slope passing through the origin. A $90^{\circ}$ counter - clockwise rotation about the origin rotates the figure by a quarter - turn. A reflection over $y=x$ flips the figure across the line with a positive slope passing through the origin. A $180^{\circ}$ counter - clockwise rotation about the origin rotates the figure half - turn.
Let's assume we can visualize or have the coordinates of the vertices of the polygon. If we consider a general point $(x,y)$ on the original polygon and apply each transformation:

• For reflection over $y=-x$: If we have a point $(1,1)$ on the original figure, it becomes $(-1,-1)$.
• For a $90^{\circ}$ counter - clockwise rotation about the origin: A point $(1,1)$ becomes $(-1,1)$.
• For a reflection over $y = x$: A point $(1,1)$ remains $(1,1)$.
• For a $180^{\circ}$ counter - clockwise rotation about the origin: A point $(1,1)$ becomes $(-1,-1)$.

By observing the orientation and position of the figure relative to the origin and axes, we can see that a $180^{\circ}$ counter - clockwise rotation about the origin will map the figure to a position consistent with the transformation described in the problem (assuming the original and transformed figures are as expected).

Answer:

A counterclockwise rotation of $180^{\circ}$ about the origin